CN112055871A - Computer-implemented method for identifying mechanical properties by coupled image correlation and mechanical modeling - Google Patents

Abstract

The general field of the invention is computer-implemented methods of identifying mechanical parameters of an object subjected to mechanical stress. The method according to the invention comprises the following steps: a step of acquiring images of the subject taken before and during the application of the mechanical stress by the imaging device; three steps of calculating the influence due to the execution of the stress on the basis of the modeling of the recorded images or on the basis of a theoretical mechanical modeling of said stress; a step of defining a functional equal to the difference between the two models; and a final step of minimizing said functional so that the experimental model is as close as possible to the theoretical mechanical model. The additional measures enable the method according to the invention to be refined.

Description

Computer-implemented method for identifying mechanical properties by coupled image correlation and mechanical modeling

The general field of the invention is that of materials and structures science, more precisely that of knowledge of the behaviour of materials and structures when subjected to various mechanical stresses.

The main industrial applications are related to the strength of the material. However, there are other fields of application such as non-destructive testing, certain biological applications or medical elastography.

Typically, the measurements are made by means of a mechanical test system comprising an imaging device. These devices are mainly video or still cameras operating at or near visible wavelengths, or microscopic analysis systems such as scanning electron microscopes or atomic force microscopes. Tomographic devices can also be used to analyze the entire volume of a material sample. Such devices are of various types. Non-exhaustive reference will be made to X-ray tomography devices, magnetic resonance tomography devices or "MRI" and optical coherence tomography devices.

The general operating principle of these imaging devices is as follows. For example, a number of images representing the sample are taken before and during the application of a determined mechanical or thermal load condition. These images may be two-dimensional, stereoscopic or video, or even volumetric images. Of course, if it is desired to identify mechanical properties having physical dimensions, knowledge of the dimensions of the sample is essential.

There are various ways in which the load effect can be characterized using the obtained image. These images can be used by means of digital image Correlation (CIN), a method also known under the acronym "DIC". Reference will be made in particular to the publication "Image correlation for shape, motion and deformation measures: basic definitions, the term and applications", MA Sutton, JJ Orteu, H Schreier, Springer (2009). The method includes resolving motion of the sample based on the representative kinematics. This basis may be a description of the "finite element" type based on a geometric mesh of the sample. The analysis of the plurality of images obtained under load in comparison with the image or images obtained under no load makes it possible to measure the displacement field U of the sample under stress_CIN(x) And x represents the coordinates of the sample point. FIG. 1 illustrates this general principle of digital image correlation method characterization.

The same mechanical experiment can be numerically modelled by means of finite element methods or by means of other techniques to calculate the subsequent U_CAL(x) The displacement field indicated.

Modeling requires:

-knowledge of the geometry of the sample;

-knowledge of load conditions, including time;

-knowledge of boundary conditions such as displacement of zone edges or force measurements;

-knowledge of the laws of mechanical behaviour of the sample or of the phases constituting the sample;

-knowledge of the exact position of the phases for a multi-phase medium.

These different elements may be referred to collectively as { p }_iThe parameter set of (j) is described mathematically, i is the index that varies from 1 to n, n is the number of parameters in question. FIG. 2 shows a table by modelingThis general principle of characterization.

Of course, only when the displacement field U is present_CAL(x) And displacement field U_CIN(x) The same or very close together, the modeling can truly reflect the observed values. Thus, the parameter { p }_iThe optimization of is achieved by means of an iterative loop in which the parameter is modified until the two displacement fields correspond to each other to the best possible extent. Fig. 3 shows the whole process of measurement, modeling and optimization, enabling the correct parameters to be determined.

This approach has several disadvantages. In particular, images are always noisy and the degree to which DIC is more or less sensitive to such noise depends on the choice of the kinematic basis. This effect is particularly pronounced at the edges of the area under investigation. The main drawback of the modeling is that the sensitivity of the displacement field to certain parameters is low, making these parameters difficult to determine. Even possible at all parameters p_iThere is a contamination effect between. Finally, the displacement field is not modifiable throughout the identification process, despite measurement defects.

In order to optimize this method, various methods have been proposed. Thus, US patent US 7257244 entitled "imaging modalities for characterization of properties" describes an iterative method of coupling between the correlation of images and the modeling of the elastic properties of the sample in question, in the defined context of Elastography, until the best coincidence between the images is obtained. In this method, the boundary conditions are assumed to be known and are not reconsidered during the iteration.

J.R Thor in a publication entitled "A full Integrated noise distribution protocol for the identification of static lamps from digital images", published in int.J. Num.meth.Eng.84:631-660 (2010). The method relates identification and DIC by introducing a functional to be minimized, which relates two targets by enforcing identification of two displacement fields. The identifying is performed in part by minimizing a deviation from equilibrium based on a quadratic functional of a second order differential operator applied to the displacement. Its minimization therefore involves a fourth order differential operator with respect to the displacement U.

The described image correlation is a global approach based on a "finite element" type discretization of the displacement field. For the same displacement field, once the current displacement field is corrected, minimizing the image quadratic difference between the reference image and the distorted image and the deviation from the balance makes it possible to find a compromise point between the measured values obtained by the DIC and the recognition results, and optimize this compromise point with respect to the target constitutive parameter { p }. The boundary condition that the transmitted force is not zero is generated directly and uniquely from the image correlation. The method provides for the identification of elastic and non-linear constitutive laws via the registration of images. However, it should be emphasized that the measurement of edge displacements is severely affected by noise, and the use of a quadratic functional based on the second derivative of the displacement U (produced by minimizing the fourth order differential operator) results in a significant increase in sensitivity to noise. The examples processed in this way clearly demonstrate that the error near the edge where the kinematics is applied is greatly increased. These boundary conditions may even prevent a correct determination of the mechanical properties in more unfavourable cases than the conditions used in the reference. Thus, it can be observed that even though the proposed DIC/identification coupling solves some of the above disadvantages, there are other weaknesses that may limit the quantitative identification capability.

More specifically, one subject of the invention is a computer-implemented method of identifying at least one mechanical parameter, called "target parameter", of an object subjected to mechanical stress, characterized in that it comprises the following steps:

-step 1: acquiring, by an imaging device, at least two images of the object taken before and during the application of the mechanical stress and measuring a scale factor of the object;

-step 2: calculating a first functional T_CIN(U_CIN) Said first functional T_CIN(U_CIN) Corresponding to the correlation of digital images depending on the displacement field U represented using a first kinematic basis_CINSaid displacement field U_CINIs under negative pressureMeasured at any point of the object under stress between the images of the object, with and without load;

-step 3: calculating a calculated displacement field U at any point of the object_CAL；

-step 4: based on the calculated displacement field U_CALCalculating a second functional T_CAL(U_CALP, q), the calculated displacement field U_CALIs expressed using a second kinematic basis, said second functional corresponding to a variational formula of a mechanical model of said stress, said variational formula depending on said geometry of said object, applied forces, boundary conditions, at least said target parameter { p } and a predetermined mechanical parameter { q };

-step 5: calculating a third functional T in the form of a quadratic norm_PAR(U_CIN,U_CAL) Said third functional T_PAR(U_CIN,U_CAL) Is equal to U_CINAnd U_CALThe difference between the two;

-step 6: general functional T_TOT(U_CIN,U_CALP, q) with respect to U_CIN、U_CALAnd { p } minimization, said overall functional T_TOT(U_CIN,U_CAL, { p }, { q }) include at least the following:

T_TOT(U_CIN,U_CAL,{p},{q})＝αT_CIN(U_CIN)+βT_CAL(U_CAL,{p},{q})+γT_PAR(U_CIN,U_CAL)

- α, β and γ are three non-zero weighting coefficients, said weighting coefficients (α, β, γ) being based on uncertainties associated with the various quantities involved in the functional and/or on the function T_TOTThe condition number that minimizes the problem of tangency is adjusted.

Advantageously, the second functional T is adapted to be used when the behavior of the object is subjected to a time-dependent stress_CAL(U_CALP, q) depends on the determined time.

Advantageously, step 1 of the method comprises an additional measurement F of force, time or temperature_MESStep 3 of the method comprisesEvaluation value F corresponding to additional measured value_CALStep 4 is followed by step 4bis, step 4bis being the calculation of the fourth functional T_FOR(F_CAL,F_MES) Fourth functional T_FOR(F_CAL,F_MES) Proportional to the second deviation between these quantities, and the overall functional T of step 5_TOT(U_CIN,U_CALP, q) equals:

T_TOT(U_CIN,U_CAL,{p},{q})＝αT_CIN(U_CIN)+βT_CAL(U_CAL,{p},{q})+γT_PAR(U_CIN,U_CAL)+χT_FOR(F_CAL({p},{q}),F_MES)

χ is a fourth weighting coefficient, which is a function of the uncertainty associated with the various quantities involved in the functional and/or of the functional T_TOTIs adjusted to minimize the condition number of the tangent problem.

Advantageously, the overall functional T_TOTIs achieved by an iterative method, which may or may not require the calculation of T_TOTOf the gradient of (c).

Advantageously, the first kinematic basis is identical to the second kinematic basis.

Advantageously, the measurement uncertainty is determined from a function T with convergence_TOTIs estimated by a measurement of the available acquisition noise.

Advantageously, the first kinematic basis or the second kinematic basis is generated on a finite element mesh.

Another subject of the invention is a computer device adapted to identify at least one mechanical parameter according to the method described above.

Another subject of the invention is a computer-readable medium having a program for executing the method according to the above-mentioned method.

The invention will be better understood and other advantages will become apparent from a reading of the following non-limiting description and from the accompanying drawings, in which:

FIG. 1, which has been described, shows the various steps of monitoring to represent stress in a sample using a digital image correlation method known as CIN (DIC);

FIG. 2, which has been described, illustrates the various steps of representing stress in a sample by a modeling method;

fig. 3, which has been described, shows the various steps of optimizing a target parameter { p } (such as mechanical properties, geometry or boundary conditions) by an iterative method according to the prior art;

fig. 4 shows the various steps of optimizing the target parameters by the iterative global method according to the invention.

By way of example, FIG. 4 illustrates the various steps of an iterative global method to optimize a target parameter in accordance with the present invention. If the method according to the invention is compared with the prior art method according to fig. 3, the main variation between these two methods is the position of the kinematic calculation, in particular with respect to the boundary conditions. In the method according to the invention, modeling also intervenes in these kinematic values, which are therefore no longer constants but form part of the optimization loop.

The computer-implemented method according to the invention for identifying at least one mechanical parameter, called "target parameter", of a material of construction of a test piece or part subjected to a known mechanical stress comprises the following steps.

The first step consists in acquiring, by means of an imaging device, digital images of the object taken before, during and (for certain applications) after the application of the mechanical stress, and measuring the scale factor of the object. In fact, if one wants to determine mechanical properties with physical dimensions, knowledge of the dimensions of the sample is essential.

The imaging means may be any device used alone or in combination such that at least one image of an object may be obtained. The images obtained by the device may be optical images obtained in various wavelength ranges well known to those skilled in the art.

Secondly, a first functional T is calculated based on the image obtained using the selected method_CIN. The first functional corresponds to a displacement field based U_CINPhase of a digital imageOf interest, the displacement field U_CINIs represented using a first relative kinematic basis (e.g., on a finite element mesh).

Traditionally, this first functional is the sum of squared differences between the reference image over the investigation region and one or more corrected distortion images of the displacement field, but other criteria such as cross-correlation or joint entropy of the information may be chosen.

The first step of the method may comprise additional measurements F, such as force, time or temperature measurements_MES。

The third step comprises calculating a calculated displacement field U using the second kinematic base representation_CALThis second functional corresponds to a variation formula of the mechanical model of the stress, said variation formula depending on the geometry of the object, the applied force, the boundary conditions, at least the target parameter { p } and the predetermined mechanical parameter { q }.

In fact, the same mechanical experiment can be numerically simulated by finite element method or by other techniques to calculate the subsequent U_CAL(x) The displacement field indicated.

The fourth step includes calculating a second functional T_CAL(U_CALP, q), the second functional T_CAL(U_CALP, q) depends on the mechanical behaviour of one or more materials, the geometry of the part, boundary conditions that may include the applied force and one or more of the times in question grouped together in the form of the target p or predetermined q parameters, and the calculation of a nominal set of values for the target parameters. The fourth step of the method may comprise comparing the additional measured values F_MESCorresponding evaluation values F for force, time or temperature (if they are available)_CALSo that an additional functional T proportional to the squared difference of the last two quantities can be formulated_FOR(F_CAL,F_MES) The additional functional T_FOR(F_CAL,F_MES) Possibly weighted by the inverse of the variance of the measured values.

The fifth step, with U_CINAnd U_CALThe quadratic norm form of the difference between them introduces a third functional T_PAR(U_CAL,U_CIN). These two displacement fields are combined with other physical quantities measured and calculated with the target parameters well identified and the predetermined parameters appropriate, and with the model used.

Therefore, the proposed recognition principle is to weight sum T of these three or four functional in the final step_TOTAbout two displacement fields U_CALAnd U_CINAnd the target parameter { p } is minimized:

T_TOT(U_CIN,U_CAL,{p},{q})＝αT_CIN(U_CIN)+βT_CAL(U_CALV,{p},{q})+γT_PAR(U_CIN,U_CAL)+χT_FOR(F_CAL({p},{q}),F_MES)

in case the method does not comprise additional measurement values, the functional T_TOTSimplified to the first three terms.

It should be noted that if the stress range, geometry, and even the definition of the image are not appropriate, the problem may still be ill-posed (ill-posed). In this case, not all target parameters may be measured. Tikhonov regularization, which corresponds to the penalty of a deviation between an identified parameter and an expected parameter, may require numerical solution to the problem. The obtained solution should then be judged using its own uncertainty, e.g. by considering the previously characterized measurement noise pair minimization T_TOTWithout considering Tikhonov regularization.

The weighting coefficients (α, β, γ, χ) enable the use of the uncertainty associated with the quantity involved in the functional and/or of the functional T_TOTThe condition number of the tangent problem is minimized to give greater or lesser importance to the terms. It should be noted that any one of the arbitrarily selected weights may be set to 1.

If the variational expressions of the mechanical model cannot be obtained directly, it should be noted that the functional T_PARAbout U_CALCan be simply represented, for example, in finite element codes by a linear elastic connection that produces a logical-to-U at each node_CALAnd U_CINIn proportion to the deviation therebetweenNodal forces. Thus, for finite element modeling performed using current state of the art computer code (which can include arbitrarily complex constitutive relations), the proposed T_TOTFormula is in its relation to U_CALOnly additional wire elastic connections need to be introduced at each node of the grid. The resulting solution is exactly in U_CINAnd { p } and { q } are fixed, the solution that minimizes the overall functional. By alternating the minimization steps with respect to different subsets of unknowns, it is possible to achieve a target minimization if the problem is adaptive.

For example, functional T may be performed by means of a Newton-Raphson method via continuous linearization and correction_TOTIs minimized.

Advantageously, the functional T has convergence_TOTThe Hessian function (Hessian) of (a) makes it possible to estimate the measurement uncertainty if, for example, a measurement of the acquisition noise is obtained via repeated acquisitions in the stress-free case before the mechanical experiment is carried out. In particular, it is understood that the fitness of the problem corresponds to strictly positive eigenvalues, and in this case, the condition number corresponds to the spectral radius of the hessian function. Otherwise, Tikhonov regularization can be proposed.

Advantageously, these elements also validate or invalidate the model. Specifically, the method comprises the following steps:

-displacement field U_CINThe residual field, i.e. U, enabling the estimation of the image correlation_CINDistortion of the displacement field and the difference between the corrected image and the reference image;

modeling makes it possible to verify whether constitutive laws and equilibrium conditions are satisfied;

-any additional measurements available are compared with those derived from the modeling;

two displacement fields (one displacement field is close to the measured value U)_CINAnother displacement field is close to the model U_CAL) Are combined in the same functional that measures the consistency of the two methods.

Thus, each functional used provides its own verification. Conversely, the residual is too large to be compatible with the acquisition noise signal model or measurement errors, and provides an indication as to how rich the explanatory model is or to identify unintended measurement artifacts.

Advantageously, the coupling of the terms of a functional enables compensation of the ill-conditioned or pathological nature of that functional or that functional. For example, low contrast or poorly illuminated areas may not allow the use of only functional T_CINTo measure U_CIN. Then, functional T_PARThe lack of information can be compensated for by calculation.

Symmetrically, the coupling functional T when constitutive laws or geometric non-linearities lead to a loss of stability or uniqueness of the solution to the mechanical problem_PARThis then allows the fitness of the problem to be recovered and the same branch to be tracked via the model.

In the case of convergence, for the displacement field U_CINDifferent calculated displacement field U_CALModel F_CALAnd the estimate of the target parameter p of the identified object, the total functional reaches its minimum value. These parameters may be parameters of the material associated with one or more facies, or the geometry or other quantities (e.g., boundary conditions) of the object.

The method is computer-implemented, so that the computer device can be adapted to recognize at least one mechanical parameter according to the method described above.

The method according to the invention can therefore be implemented substantially by means of a numerical calculation device that is fully implementable using current calculation tools, but in practice a wider range of materials can be analysed, or cheaper hardware used (in particular in relation to the quality of the acquisition hardware given the end result).

Claims (10)

1. A computer-implemented method of identifying at least one mechanical parameter, called "target parameter", of an object subjected to mechanical stress, characterized in that it comprises the following steps:

-step 1: acquiring, by an imaging device, at least two images of the object taken before and during the application of the mechanical stress and measuring a scale factor of the object;

-step 2: calculating a first functional T_CIN(U_CIN) Said first functional T_CIN(U_CIN) Corresponding to the correlation of the digital images depending on the displacement field U represented using the first kinematic basis_CINSaid displacement field U_CINIs measured between the images of the object at any point of the object under stress, with and without load;

-step 3: calculating a calculated displacement field U at any point of the object_CAL；

-step 4: based on the calculated displacement field U_CALCalculating a second functional T_CAL(U_CALP, q), the calculated displacement field U_CALIs expressed using a second kinematic basis, said second functional corresponding to a variational formula of a mechanical model of said stress, said variational formula depending on the geometry of said object, the applied force, boundary conditions, at least a target parameter { p } and a predetermined mechanical parameter { q };

-step 5: calculating a third functional T in the form of a quadratic norm_PAR(U_CIN,U_CAL) Said third functional T_PAR(U_CIN,U_CAL) Is equal to U_CINAnd U_CALThe difference between the two;

-step 6: general functional T_TOT(U_CIN,U_CALP, q) with respect to U_CIN、U_CALAnd { p } minimization, said overall functional T_TOT(U_CIN,U_CAL, { p }, { q }) include at least the following:

T_TOT(U_CIN,U_CAL,{p},{q})＝αT_CIN(U_CIN)+βT_CAL(U_CAL,{p},{q})+γT_PAR(U_CIN,U_CAL)

α, β and γ are three non-zero weight coefficients.

2. The computer-implemented method of identifying at least one mechanical parameter of claim 1, wherein the object is a human beingIs subjected to time-dependent stress, the second functional T_CAL(U_CALP, q) depends on the determined time.

3. Computer-implemented method of identifying at least one mechanical parameter according to any of the preceding claims, characterized in that step 1 of the method comprises an additional measurement F of force, time or temperature_MESStep 3 of the method comprises an evaluation value F corresponding to the additional measured value_CALStep 4 is followed by step 4bis, said step 4bis being the calculation of a fourth functional T_FOR(F_CAL,F_MES) Said fourth functional T_FOR(F_CAL,F_MES) Proportional to the second deviation between these quantities, and the overall functional T in step 5_TOT(U_CIN,U_CALP, q) equals:

T_TOT(U_CIN,U_CAL,{p},{q})＝αT_CIN(U_CIN)+βT_CAL(U_CAL,{p},{q})+γT_PAR(U_CIN,U_CAL)+χT_FOR(F_CAL({p},{q}),F_MES)

χ is a fourth weighting coefficient, according to the uncertainty associated with the various quantities involved in the functional and/or according to the functional T_TOTIs adjusted to minimize the condition number of the tangent problem.

4. The computer-implemented method of identifying at least one mechanical parameter according to any of the preceding claims, wherein the overall functional T_TOTIs achieved by an iterative method.

5. Computer-implemented method of identifying at least one mechanical parameter according to any one of claims 1 to 3, characterized in that said overall functional T_TOTIs achieved by an iterative method, requiring the calculation of T_TOTOf the gradient of (c).

6. The computer-implemented method of identifying at least one mechanical parameter of any of the preceding claims, wherein the first kinematic basis is the same as the second kinematic basis.

7. The computer-implemented method of identifying at least one mechanical parameter according to any of the preceding claims, characterized in that a measurement uncertainty is determined by said functional T having a convergence_TOTIs estimated by a measurement of the available acquisition noise.

8. The computer-implemented method of identifying at least one mechanical parameter according to any of the preceding claims, characterized in that the first kinematic basis or the second kinematic basis is generated on a finite element mesh.

9. A computer device adapted to identify at least one mechanical parameter as claimed in any one of claims 1 to 8.

10. A computer readable medium having a program for performing the method of any one of claims 1 to 8.

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